A generalized lower‐bound conjecture for simplicial polytopes

Let P be a simplicial d‐polytope, and, for – 1 ≤ j < d , let fj ( P ) denote the number of j ‐faces of P (with f _1 ( P ) = 1). For k = 0, …, [½d] – 1, we defineg k     ( d + 1 )( P ) = ∑ j = − 1 k( − 1 )k − j(d − jd − k) f j ( P ) ,and conjecture thatg k   ( d + 1 )( P ) ⩾ 0 ,with equality in the k ‐th relation if and only if P can be subdivided into a simplicial complex, all of whose simplices of dimension at most d – k – 1 are faces of P . This conjecture is compared with the usual lower‐bound conjecture, evidence in support of the conjecture is given, and it is proved that any linear inequality satisfied by the numbers fj (P) is a consequence of the linear inequalities given above.